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I would like to make a cross section like this (only the hatched part)

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(image source)

using bézier curves/circles, to then use in a screw modifier and make a drill bit.

I initially thought I could use a Boolean operation (difference) with a set of Bézier circles like this:

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but for Béziers there seem to be no Boolean modifiers. When I convert them to mesh (using Alt-C), the Boolean modifier becomes available but "difference" has no effect.

Apart from manually shaping a curve (by dragging nodes and handles), is there another, more "mathematical" way, of achieving the result?

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Use Boolean modifier after adding some thickness to circles so modifier to work or use Knife Project.

Cutting with Boolean modifier

Boolean didn't work in your case most likely because there wasn't geometry needed for it to operate with. Two flat faces won't be cut with any Boolean operation; if you extrude these faces in any direction (e.g. by Z axis) and then add the Boolean, it will be executed correctly.

  1. First fill the circles with geometry: if they are still curves convert them to 2D; if they are meshes select all the vertices and press F. Position them accordingly: center one should be as separate object, and 3 others should be joined as another one.

  2. Extrude them / add Solidify modifier; you may leave small values in order the extrusion to be deleted further on as double vertices (increase threshold a bit to achieve that).

  3. Finally add Boolean to the center circle and choose object with 3 circles as a target. Delete created additional geometry if needed.

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Knife Project

It's basically described here. In this case you shouldn't extrude anything. Some general steps:

  1. Place circles as in the 1-st way;
  2. Select those 3 joined as the first object, then one in the center and enter their Edit mode (the selection order is important because it determines where faces will be cut from).
  3. Press Numpad 7 to enter Top View. It will make Knife Project to cut object using this projection.
  4. In the Toolshelf (open with T) find Knife Project and choose it.

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See also:

Cases where boolean operations can fail

How would you cut a shape out of an object using another object?

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  • $\begingroup$ Great, thank you! A few comments after experimenting: 1) On Cutting with Boolean modifier - step 1: converting bézier circles to 2D is not enough it seems (I can't extrude/solidify them in the next step), they need to be converted to mesh (Alt-C -> Mesh from Curve). - by using a Solidify modifier in step 2, it's easy to convert the result of step 3 to a bézier curve again. Just apply the Boolean modifier, remove the Solidify modifier, and do Alt-C -> Curve from Mesh. (Optionally use the "Simplify Curves" add-on to finish.) $\endgroup$ – Abel Jan 18 '16 at 14:58
  • $\begingroup$ 2) On Knife project: I found it works best with - the middle circle (the object to be cut) being a bézier circle converted to mesh (with Alt-C) and filled (Edit mode, select all, F), but not extruded/solidified - and the other three circles simply three bézier circles joined into one object (as bézier circles), without filling/extruding/.... $\endgroup$ – Abel Jan 18 '16 at 14:58
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    $\begingroup$ @Abel 1. of course you should convert curves to mesh first if extrude needed as it's done on faces in this case which are present in case of mesh. Sorry for not mentioning that 2. I mentioned that no extrusion is needed if Knife Project is used (it still should work though). I didn't notice any changes in its work regardless of whether 3 other circles are curves / meshes and filled / unfilled. So these are tiny details which imo make difference only in workflow (but may be important in terms of convenience). $\endgroup$ – Mr Zak Jan 18 '16 at 15:18

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